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Slack variables give an embedding of a polytope into the standard f-orthant, where is the number of constraints (facets of the polytope). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized), and is expressed in terms of the constraints (linear functionals, covectors).
with v the Lagrange multipliers on the non-negativity constraints, λ the multipliers on the inequality constraints, and s the slack variables for the inequality constraints. The fourth condition derives from the complementarity of each group of variables (x, s) with its set of KKT vectors (optimal Lagrange multipliers) being (v, λ). In that case,
For any greater-than constraints, introduce surplus s i and artificial variables a i (as shown below). Choose a large positive Value M and introduce a term in the objective of the form −M multiplying the artificial variables. For less-than or equal constraints, introduce slack variables s i so that all constraints are equalities.
So if the i-th slack variable of the primal is not zero, then the i-th variable of the dual is equal to zero. Likewise, if the j-th slack variable of the dual is not zero, then the j-th variable of the primal is equal to zero. This necessary condition for optimality conveys a fairly simple economic principle.
Theory of two-level planning; Variable splitting; Basic solution (linear programming) — solution at vertex of feasible region; Fourier–Motzkin elimination; Hilbert basis (linear programming) — set of integer vectors in a convex cone which generate all integer vectors in the cone; LP-type problem; Linear inequality
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The variable for this column is now a basic variable, replacing the variable which corresponded to the r-th column of the identity matrix before the operation. In effect, the variable corresponding to the pivot column enters the set of basic variables and is called the entering variable , and the variable being replaced leaves the set of basic ...
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